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Abstract

Let ${\cal M}$ be a type II$_1$ von Neumann algebra, $\tau $ a trace in ${\cal M}$, and $L^2({\cal M},\tau )$ the GNS Hilbert space of $\tau $. If $L^2({\cal M},\tau )_+$ is the completion of the set ${\cal M}_{\rm sa}$ of selfadjoint elements, then each element $\xi \in L^2({\cal M},\tau )_+$ gives rise to a selfadjoint unbounded operator $L_\xi $ on $L^2({\cal M},\tau )$. In this note we show that the exponential $\mathop {\rm exp}\nolimits :L^2({\cal M},\tau )_+ \to L^2({\cal M},\tau )$, $\mathop {\rm exp}\nolimits (\xi )=e^{iL_\xi }$, is continuous but not differentiable. The same holds for the Cayley transform $C(\xi )=(L_\xi -i)(L_\xi +i)^{-1}$. We also show that the unitary group $U_{\cal M}\subset L^2({\cal M},\tau )$ with the strong operator topology is not an embedded submanifold of $L^2({\cal M},\tau )$, in any way which makes the product $(u,w) \mapsto uw$ ($u,w\in U_{\cal M}$) a differentiable map.